Marcinkiewicz--Zygmund inequalities for scattered and random data on the $q$-sphere

Abstract

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere ${\mathbb{S}}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere ${\mathbb{S}}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points $\hat{I}{¾}_1,\dots ,\hat{I}{¾}_N$ on ${\mathbb{S}}^q$, the dimension $q$, and the polynomial degree $n$.